文档介绍：Kreisel and Lawvere on
Category Theory and the
Foundations of Mathematics
Jean-Pierre Marquis
Université de Montréal
Montréal Canada
Impact of categories
• When the impact of categories on
foundations is discussed with
« mainstream » logicians, we often get
two responses:
1. Pragmatic scepticism: still waiting for new
significant results;
2. Philosophically motivated objections.
Claims
1. Kreisel has articulated a view about the foundations
of mathematics and category theory that prevails
among logicians even today;
2. Although this view had some credibility and force
when it was formulated, it ought to be reevaluated;
3. Kreisel’s view is based on certain assumptions
which are dubitable and ought to be contrasted with
alternatives, in particular with Lawvere’s views.
Kreisel’s claims: the sources
1. Appendix to Feferman’s paper on the
foundations of category theory in 1969;
2. « Observations on popular discussions of
foundations » in Axiomatic Set Theory
1971;
3. A review of Mac Lane’s « Categorical
algebra and set-theoretic foundations » in
Axiomatic Set Theory 1971;
4. Appendix to Elements of Mathematical
Logic with Krivine 1967.
The socio-historical context
1. « Mainstream » developments and
research programs in the foundations
of mathematics in the 1960’s;
2. Category theory in the 1960’s.
1963: Berkeley
• Meeting on model theory:
– It is the who’s who of logic and the
foundations of mathematics.
1963
• We can be category theorists:
– Lecture on categorical algebra at the
AMS;
– Grothendieck’s SGA4;
– Freyd’s presentation of his AFT;
– Lawvere’s thesis;
– Ehresmann’s « catégories
structurées »;
– First coherence theorem;
– Adjoint functors and limits
1963: Lawvere’s thesis
• Algebraic categories
and algebraic functors;
• The category of
categories as a
foundation for
mathematics;
• Sets within categories;
• Central role to adjoint
fu